To solve for \( \log_2 16 \), we first express 16 in terms of base 2:
\[ 16 = 2^4 \]
Thus,
\[ \log_2 16 = \log_2 (2^4) = 4 \]
Now, we will evaluate each of the options provided using logarithm properties.
Option #1: \( 3 \log 6 \)
We know that \( \log 6 \) is not simplifiable to be equivalent to a numeric value like 4 directly. So, we will compute \( 3 \log 6 \) to see if it equals 4.
Option #2: \( \log 6 \cdot \log 6 \cdot \log 6 = (\log 6)^3 \)
This also doesn’t evaluate directly to a numerical value of 4, similar reasoning as for option #1.
Option #3: \( 36 \log 6 \)
Again, we cannot directly determine whether this is equal to 4 without knowing \( \log 6 \).
To find which of the options is equivalent to \( \log_2 16 = 4 \), we need to express the options in terms of a common logarithm. If we let \( \log = \log_{10} \) (common logarithm), we can use the change of base formula:
\[ \log_2 16 = \frac{\log 16}{\log 2} \]
Since we have established that \( \log_2 16 = 4 \), we equate:
\[ 4 = \frac{\log 16}{\log 2} \]
Now \( \log 16 = \log 2^4 = 4 \log 2 \). Substituting back, we have:
\[ 4 = \frac{4 \log 2}{\log 2} \implies 4 = 4 \]
Now converting each option into terms of \( \log 2 \):
- For Option #1: \( 3 \log 6 \)
- For Option #2: \( (\log 6)^3 \)
- For Option #3: \( 36 \log 6 \)
None of these options appear directly related to \( \log_2 16 = 4 \) without evaluating \( \log 6 \) in a particular base.
Thus, comparing dimensions is complex without additional information, but we need to find potential correlations through numerical investigation or transforming through additional logarithmic properties directly correlating to \( \log_2 16 \).
Given information about potential numerical equality across correlated logarithmic values could relate \( \log 6 \) to known values around \( 2 - 3 \), but it’s not definitive from each step.
Finally, checking only empirical evaluations of \( 3 \log 6 = 3(0.778) \approx 2.334 \), showing Option #1 stably falls below required equivalency.
Final Verdict:
Based on transfers back to 4 and mathematic evaluations without circular dependencies,
None of the options are equivalent to \( \log_2 16 \). Each would deviate under valued logarithm calculations; thus,
Correct conclusion: None of the supplied options are equivalent directly to \( 4 \) or \( \log_2 16 \).